We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis ($\mathsf{ETH}$), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem ($\mathsf{MLD}$) and the parameterized Nearest Codeword Problem ($\mathsf{NCP}$) within some fixed constant factor. Our starting point is the ETH-based exponential-time hardness of $(c,s)$-Gap-$\mathsf{MAXLIN}$ established in [BHI+24]. We transform a $(c,s)$-Gap-$\mathsf{MAXLIN}$ instance into an instance of $γ$-Gap $k$-$\mathsf{MLD}$ via a novel combinatorial object that we call a cover family. We provide both a randomized construction of the required cover families and a subsequent derandomization. Prior to our work, $n^{Ω(k)}$ hardness for constant-factor approximation was only shown under the randomized Gap Exponential Time Hypothesis Gap-$\mathsf{ETH}$ [Man20], which is a much stronger assumption than $\mathsf{ETH}$. Under $\mathsf{ETH}$, the strongest known lower bound was $n^{Ω(k/\operatorname{poly} \log k)}$ due to [BKM25]. Unlike previous approaches that rely on reductions from the hardness of approximating $2$-$\mathsf{CSP}$, our reduction provides a more direct and conceptually simpler route to achieving the optimal lower bounds.

翻译：我们提出一种简单的确定性归约，在指数时间假设（$\mathsf{ETH}$）下，该归约给出了在某个固定常数因子内逼近参数化最大似然解码问题（$\mathsf{MLD}$）和参数化最近码字问题（$\mathsf{NCP}$）的紧致下界。我们的出发点源于[BHI+24]中建立的基于ETH的$(c,s)$-Gap-$\mathsf{MAXLIN}$指数时间困难性。通过一种称为覆盖族的新型组合对象，我们将$(c,s)$-Gap-$\mathsf{MAXLIN}$实例转化为$γ$-Gap $k$-$\mathsf{MLD}$实例。我们提供了所需覆盖族的随机化构造及其后续去随机化方法。在我们之前，常数因子逼近的$n^{Ω(k)}$困难性仅在随机化间隙指数时间假设（Gap-$\mathsf{ETH}$）[Man20]下得到证明，而该假设远强于$\mathsf{ETH}$。在$\mathsf{ETH}$下，此前已知的最强下界来自[BKM25]的$n^{Ω(k/\operatorname{poly} \log k)}$。与依赖$2$-$\mathsf{CSP}$逼近困难性的归约不同，我们的归约为实现最优下界提供了一条更直接且概念上更简洁的途径。